Four Wave Mixing in Silicon Photonics - Optical Signal Processing by Silicon Photonics - page 107

evolution of the pump and signal envelopes inside the SOI waveguide as a result
of FWM parametric process. It can result in amplification of the signal as well
as generation of an idler wave ( ω i ) at the frequency ω i = 2 ω p − ω s [ 24 , 25 ].

Assuming that | Ap | > | As | > | Ai |

γ p + i β
2

dA p
dz =−

1
2

2

α + α FC A
p

( z ) A p + i A p A p

(8.3)

dA s
dz =−

1
2

γ s + i β
2

2

α + α FC A
s

A s + i γ s A p A i exp (− i ∆ k · z ) ( z ) A s + 2 i A p

(8.4)

dA i
dz =−

1
2

γ i + i β
2

A i − i γ i A 2 p

2

α + α FC A
i

A i − 2 i

( z ) A p A s exp ( i ∆ k · z )

(8.5)

The first term on right-hand side of Eqs. ( 8.3 - 8.5 ) is responsible for attenuations
due to linear absorption and free-carrier absorption, the second term is respon-
sible for SPM and TPA in Eq. ( 8.3 ) and XPM and TPA in Eqs. ( 8.4 ) and ( 8.5 ),
and the last term in Eqs. ( 8.4 ) and ( 8.5 ) describes the energy transfer between
the interacting waves. The use of linear loss coefficient α = 1.4 dB/cm the TPA
coefficient β = 0.75 × 10 − 11 m / W and the nonlinearity coefficient γ j = n 2 ω p / c
with the nonlinear refractive index n 2 = 5.5 × 10 − 18 m 2 / W ( j = p , s , i ) . ∆ k is
the phase mismatch due to propagation constants. The TPA-induced FCA loss is
given as α FCA
j

2 N , where λ j is the wavelength ( nm ) ,

( z ) = 1.45 × 10 17 λ j / 1,550

cm − 1 is carrier density generated by TPA. Here N should satisfy the following

rate equation at any position of the waveguide at any time [ 26 ].

N

dN ( t , z )
dt

β
2 hv I 2 ( t , z ) −

N ( t , z )
τ

(8.6)

=

In this equation, I is the peak intensity, hv is the photon energy, and τ is the effective
carrier lifetime, which changes with the waveguide geometry or reverse bias voltage
if a p-i-n diode structure exists. For CW pumping or long pulse pumping, N will
reach the local steady-state value of N ( z ) = τβ I 2 ( z )/ hv . For pulse pumping, the
repetition rate R of the pulsed pump is an important factor impacting N . Under the
operating condition of pulse pumping with pulse width T 0 << τ , N is given by

1
1 − e − 1 / R τ

β T 0 I 2 ( t , z )
2 hv

N ( t , z ) ≈ (8.7)

FWM is a coherent process whose efficiency depends on how well the phase mis-
match meets the phase matching condition

k = k s + k i + 2 k p + 2 γ P p
= n s ω s + n i ω i − 2 n p ω p c + 2 γ P p

(8.8)

2

= β 2 ω s − ω p + 2 γ P p